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Differentiability
Those friendly functions that don't contain breaks, bends or cusps are "differentiable". Take their derivative, or just infer some facts about them from the Mean Value Theorem.
\[\lambda=\left( \frac{f(5102)-f(2015)}{f'(c)} \right) \left( \frac{f^2(2015)+f^2(5102)+f(2015)f(5102)}{f^2(c)} \right)\]
Let \(f:[2015,5102] \rightarrow [0,\infty)\)be any continuous and differentiable function. Find the value of \(\lambda\), such that there exists some \(c\in [2015,5102]\) which satisfies the equation above.
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by
Sandeep Bhardwaj
\[ f(x) = \displaystyle \prod_{k=1}^{\infty} \left( \dfrac{1+ 2\cos\left( \dfrac{2x}{3^k} \right)}{3}\right) \]
Let \(f(x)\) denote an function as described above. The number of points where \( |xf(x)| + | |x-2|-1| \) is non-differentiable in \( x \in (0,3\pi) \) is \(k\), find \(k^2\).
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by
Upanshu Gupta
such that the tangents at these points to \(S\) are perpendicular to the line \[l \ : \ (\sqrt {2}-1)x+y=0.\]
Find the value of \[\left\lfloor \displaystyle\sum_{k=1}^{n} |\alpha_k|-|\beta_k|\right\rfloor.\]
Details and Assumptions
\(\lfloor{\cdots}\rfloor\) denotes the floor function (greatest integer function).
Even though in the problem, a plural form of points is given, there may only be one such point that exists.
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by
Pratik Shastri
What is the largest possible number of integers \(n\) such that for some fixed polynomial \(f(x) \) of degree \(3\) with integer coefficients, \(f(n)=n^{1000}\)?
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by
Calvin Lin
Let \(f(x)\) be a non-constant thrice differentiable function defined on real numbers such that \(f(x)=f(6-x)\) and \(f'(0)=0=f'(2)=f'(5)\). Find the minimum number of values of \(p \in [0,6]\) which satisfy the equation \[(f''(p))^2+f'(p)f'''(p)=0\] Details and Assumptions:
\(f'(p)=\left( \frac{df(x)}{dx} \right)_{x=p}\)
\(f''(p)=\left( \frac{d^2f(x)}{dx^2} \right)_{x=p}\)
\(f'''(p)=\left( \frac{d^3f(x)}{dx^3} \right)_{x=p}\)
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by
Sandeep Bhardwaj